From the 2nd "So" to the 5th "So" are just different ways of writing 0 = 0 and the steps are all perfectly valid. But when you divide through by (a-b), you are dividing by nought and it all falls apart.

x/0 has no value whatsoever, because there is NO number (at least not in the real world, where most of us live) which when multiplied by 0 will give you x.

The expression is therefore undefined and has no meaning. Trying to prove that it has a value - be that a single value or an infinite number of values - is pointless. (Though I'm guessing that there is some theoretical mathematician out there that is trying to prove me wrong.)

If you do want to prove it to yourselves, try plotting a graph of y=1/x. You'll find that the closer and closer x gets to 0, the closer y gets to infinity. But, and this is the important bit, the curve never touches the y-axis, which it would have to do in order to give a value to 1/0. So x/0 can never, ever, have any value. QED

If that were so, it is still the case that infinity has no value, therefore x/0 has no value. Theoretically, infinity is a point somewhere in the far, far distance on (at the end of) the y-axis. But because the 1/x curve never touches the y-axis. x/0 can never be infinity either.

And for the same reason the cone has an infinite surface area as well as infinite volume - the base of the cone gets wider and wider as x approaches 0 but never stops (it would stop only when it touched the y-axis at that theoretical point called infinity. Which would paradoxically result in a finite volume and finite surface area.)

... ∞ does in fact represent a number, albeit infinitely large. So, tell me, what is the result of ∞ x 0? There should be no need to ask a mathematician - it doesn't matter how big ∞ is, the answer has to be 0. That being the case, x/0 cannot be ∞.

A mathematician will tell you that x/0 is undefined - it has no value.

I spent several years at university doing proof theory. Since I've managed to prove that x/0 cannot be ∞, the onus is now on you to prove that it is.

And for the pedants amongst us, "The love of money is the root of all evil."

Saying you have proved your point then quoting your GCSE in maths or whatever does not mean your point is proven. Not to mention the fact that you missed my very interesting point earlier.

You've misunderstood the meaning of 'proof' in proof theory. I have demonstrated using logical argument that x/0 cannot equal infinity (or, to be precise, an infinitely large number). All I've done is invite others to counter that with similar logical argument. Simply saying that x and 0 are both numbers doesn't count. (But if one persists witht that argument, there is a logical counter to it.)

As for your 'interesting' point, which one? Or did you miss my response at 21:02 yesterday?

And as for Wikipedia, that doesn't surprise me. My argument is the most basic and well-known proof of why x/0 does not compute.

If you really want to understand the maths around infinity and dividing by zero, get yourselves a good book on linear analysis. You'll find that lots of apparently sound functions have a discontinuity where a graph would cross one of the axes. Mathematicians can spend their entire career studying these discontinuities.

In the real world (that most of us live in!) something divided by zero is undefined - it has no meaning. You could call it infinity or whatever else you want, but that something would not be a number like all the other 'real' numbers so you can't do the usual arithmetic (add, multiply, divide, subtract) with it. When we try to do arithmetic with it, we end up with those odd results.

What you've proven is that you have a closed mind and aren't willing to accept anything outside your own understanding.

You, Triggle and Kdauda are confusing "undefined" with "I don't know the answer to that one, my calculator won't do it, and I don't do abstract concepts".

You are correct that ∞ x 0 = 0, but that's just one of the possible values, as I'll show you.

The problem, as has been identified, is that if you introduce a division by zero into an equality, the equality breaks. We don't want an equality to break; equality's very important, particularly in the workplace.

What we need is an expression to replace the division by zero, so that the equality doesn't break.

Since you people that "did maths at university" will, I am sure, be comfortable with the fact that:

1 ÷ 0.01 = 100, and

1 ÷ 0.001 = 1,000,

and so on, such that the more you tend towards zero the larger the result gets.

It follows that when you reach zero (which is what you do when you get to 1 ÷ 0, which is what we're talking about), the number is incomprehensibly large; it's larger than any number we know; there can't be any larger number.

Let's call this incomrehensibly large number (of which there can't be any larger number) ∞ for arguments sake. It's the inverse of 0.

Now because there cant be any larger number than ∞, it follows that:

2 x ∞ = ∞, and

3 x ∞ = ∞, and

4 x ∞ = ∞,

and so on.

Already ∞ breaks the conventional rules of multiplication. So why should those rules not also break when you multiply ∞ by zero.

We already know that ∞ can be expressed in many forms using numbers that we do know:

0/0, 1/0, 2/0, 3/0, and so on until we reach ∞/0.

When we multiply by zero, the denominator in all the above fractions, we're left with the numerators, 0, 1, 2, 3, and so on until we get to ∞.

So we must conclude, must we not, that 0 ≤ (0 x ∞) ≤ ∞.

So when mikewhit says:

2 x 0 = 3 x 0 and the says "now divide by zero", we can all shout "you can't divide by zero, but you could multiply by the inverse of zero!".

I remember my physics teacher astounding the class by telling us there are many infinities, as there are an infinite number of even numbers, and even numbers represent half of all numbers, so there is a number twice of infinity (which is also conveniently called infinity). That about sums up what I learned in physics. Oh, and motorbikes don't generate enough grip to stay on the road (he proved this on the board knowing full well the method of transport I was using to go home), and yet they somehow do! I've since been told countless times never to try and prove a motorcycle can lean at high speeds and maintain grip, nor that a plane can actually stay in the air.

Btw, does anyone remember when this topic was about correct a simple mistake in an Excel forumula?

Since you people that "did maths at university" will, I am sure, be comfortable with the fact that:

1 ÷ 0.01 = 100, and

1 ÷ 0.001 = 1,000,

and so on, such that the more you tend towards zero the larger the result gets.

It follows that when you reach zero (which is what you do when you get to 1 ÷ 0, which is what we're talking about), the number is incomprehensibly large; it's larger than any number we know; there can't be any larger number.

Unfortunately the problem is that you could argue similarly with -0.01, -0.001, and so on, whereby the numbers get smaller and smaller on their way to - infinity.

Hence there isn't a uniquely defined value for 1/0. The value can't depend on which direction you approach zero from.

There are number systems where this works eg in spherical geometry 0 = 1/infinity, to visualise this you can think of the North pole as infinity and the South pole as zero.

The other problem is that your statement "there can't be any larger number" is actually wrong. There are several sizes of infinity (and indeed several ways of comparing them). I'm sure Wikepedia explains this if you look up power sets, aleph 0, and the continuum hypothesis. That's why one can prove there are more 'real' numbers than 'rationals' using an argument known as Cantor's diagonal slash.

Sort of agree with you but I did say in the 'real' world. I know I didn't define what I meant by the 'real' world (haven't got time), but one condition for the real world would certainly be that you have to be able to do conventional multiplication. But what you were doing is not in that 'real' world that I was referring to. Actually you can do arithmetic with different levels of infinities... again, not in the 'real' world.

What we need is an expression to replace the division by zero, so that the equality doesn't break.

That, I think you will find, is called moving the goalposts. As one gets closer and closer to zero, the result does become infinitely greater and greater. But we're not talking about the result when the denominator is as close to zero as we can get. We're talking about the result when it is zero. 0 may be a number, but it is a number with special properties, used to denote null, nil, nothing. 0 is not the inverse of ∞, because 0 is finite. The inverse of ∞ is 1/∞. An infinitesimally small number - close to zero, but not zero.

As Neil says, you can get as close to the y-axis as you want - but as soon as you touch it, the whole thing falls apart.

∞ x 0 can only ever have one result, not because of the properties of ∞, but because of the properties of 0. ∞ doesn't have any special numerical property - it is just a number tht happens to be so large that no-one can write it down.

So why should those rules not break when you multiply ∞ by zero

Because they don't. Although I accept that your 'proof' is elegant, it is based on fallacies. For instance, If we assume that ∞ is the largest number there is, there is no such thing as 2 x ∞, 3 x ∞ etc. But ∞ x 0 does have a value, a single value.

Also - When we multiply by zero, the denominator in all the above fractions, we're left with the numerators, 0, 1, 2, 3, and so on until we get to ∞.

Er... no we're not. Multiply anything by zero and we're left with zero. You're argument works only if zero is treated the same as any other number (including ∞). But zero is not the same as any other number.

So we each have our 'proof'. I'll defer to the real mathematicians, though. I do not know a single one, including those that I studied with, that when asked the question "what does x/0 equal?" will respond with anything other than "it doesn't equal anything".